Algebraic expressions

In an expression, terms are separated by plus (+) or minus (−) signs, and a minus sign belongs to the term that follows it. Therefore, the terms of 7x-4y+9 are 7x, -4y, and 9. Option C is incorrect because it omits the negative sign of 4y. Exam tip: Always include the sign when listing terms of an expression.

In \(p+8\), the term \(p\) can be written as \(1p\). Therefore, the coefficient of \(p\) is \(1\). The number \(8\) is the constant term, not the coefficient of \(p\). Exam tip: When no number is written before a variable, its coefficient is \(1\).

The expression has three separate terms: \(7x^2\), \(-3x\), and \(5\), so it is a trinomial. A binomial has exactly two terms. Exam tip: count terms separated by + or − signs.

Like terms must have exactly the same variables raised to exactly the same powers; only their coefficients may differ. Here, x^2 has exponent 2, whereas x can be written as x^1 and has exponent 1. Therefore, they are not like terms. Option A is a close distractor because merely containing x does not make two terms like terms. Exam tip: If no exponent is shown on a variable, treat its exponent as 1.

Any number or variable multiplied by 0 becomes 0, so \(0x=0\). Therefore, \(0x+9=0+9=9\), making \(9\) the correct option. \(9x\) is incorrect because 9 is not multiplied by \(x\). Exam tip: In simplification, a term with coefficient 0 can be removed.

By the zero multiplication property, multiplying any number or algebraic expression by 0 gives 0. Therefore, \(x\cdot0=0\). The value would be \(x\) if it were multiplied by 1, since \(x\cdot1=x\). Exam tip: If any factor in a product is 0, the entire product is 0.

By the multiplicative identity property, multiplying a variable by 1 does not change it: \(1\cdot a=a\). Therefore, \(1\cdot a+5=a+5\). \(6a\) is not correct because \(a\) and 5 are unlike terms and cannot be added. Exam tip: first simplify any multiplication by 1 in an expression.

A square has four equal sides. If each side is \(x\), its perimeter is \(x+x+x+x=4x\), so option B is correct. \(x^2\) represents the area of the square, not its perimeter. Exam tip: remember that the perimeter of a square is \(4 \times\) side.

The perimeter of a rectangle is the total length around its boundary. A rectangle has two equal lengths and two equal breadths, so its perimeter is found by adding l+3+l+3. This can also be written as twice the sum of one length and one breadth: P=2(l+3). The symbol l is an algebraic variable, so it should remain in the expression.

Substituting the given dimensions directly gives P=2(l+3)=2l+6. Option A gives only one length plus one breadth and is therefore half the perimeter. Option B is an area-like product and does not represent a boundary length, while option C has an unsuitable squared term. Hence option D, 2(l+3), is the correct expression.

The price of one pen is \(p\) rupees. Therefore, the total price of 5 pens is \(5\times p=5p\) rupees. \(p+5\) represents adding 5 rupees to the price, whereas here the price per pen must be multiplied by 5. Exam tip: Total cost of identical items = cost per item × number of items.

To find an age after 4 years, add 4 to the present age. Therefore, Riya’s age will be \(a+4\) years. The expression \(a-4\) represents her age 4 years earlier, so it is incorrect. Exam tip: use addition for “after” and subtraction for “before.”

First, the sum of \(x\) and \(3\) is \(x+3\). To take twice this entire sum, multiply the bracketed expression by \(2\): \(2(x+3)\). In \(2x+3\), only \(x\) has been doubled, so it is not correct. Exam tip: When you see “twice the sum,” put the complete sum inside brackets.

The difference of 10 and m means subtract m from 10. Therefore, the expression is \(10-m\). The expression \(m-10\) means subtracting 10 from m, so it reverses the order. Exam tip: “the difference of a and b” is generally written as \(a-b\).

The coefficient is the numerical factor multiplying the specified variable term. Here, in the term 6x^2, the factor multiplying x^2 is 6; therefore, the coefficient of x^2 is 6. The number 1 is the constant term, while x is a term with a different power. Exam tip: First identify the term with exactly the variable and exponent asked for.

A constant term is a term that contains no variable. The expression x² − x has two visible terms: x² and −x. Both contain x, so neither is constant. When a polynomial has no separately written number-only term, its constant term is taken as 0. Therefore option C is correct. Option A is the quadratic term, and option B represents the variable term with coefficient −1; neither is a constant. Option D would be correct only if a standalone 1 were actually present, but no such term appears in the given expression.

\(6a\), \(2a^2\), and \(-5\) are three separate terms because they are separated by + or − signs. Hence it is a trinomial. Exam tip: combine only like terms before counting terms.

In the expression ab + 2, a and b are two distinct variables, so option D is correct. The expressions 9x + 2 and m² + 3 each have only one variable, while 7 is a constant. Exam tip: Count the distinct letters in an expression to identify the number of variables.

In \(x^2\), the variable \(x\) has power 2 because 2 is written as its exponent. In \(2x\), 2 is the coefficient of \(x\), not its power; the power of \(x\) there is 1. In exams, look for the number written as a superscript on the variable.

An algebraic expression can contain numbers, variables, and operations. Its terms are the separate parts that are added or subtracted. For example, in an expression such as \(3x^2-5x+7\), the terms are \(3x^2\), \(-5x\), and \(+7\). The plus or minus sign belongs with the term that follows it, so the middle term is properly read as negative 5x.

Multiplication signs may occur inside a term, as in \(4xy\), but they do not usually separate terms. Division signs also do not separate terms in the usual identification of terms, and an equal sign compares two expressions rather than dividing one expression into terms. Therefore addition and subtraction are the main signs used to separate terms. Option C is correct, and the supplied explanation appropriately reminds us to keep the sign attached to its term.

Combining like terms gives \(3x-2x=x\), and combining the constants gives \(4+1=5\). Therefore, the simplified expression is \(x+5\). In \(x+3\), the constant terms have been added incorrectly. Exam tip: Add or subtract only like terms, such as \(x\)-terms with \(x\)-terms and constants with constants.